Lesson 19: Substituting to Evaluate Addition and ...

NYS COMMON CORE MATHEMATICS CURRICULUM 6•4 Lesson 19

Lesson 19: Substituting to Evaluate Addition and Subtraction Expressions

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Lesson 19: Substituting to Evaluate Addition and

Subtraction Expressions

Student Outcomes

Students develop expressions involving addition and subtraction from real-world problems.

Students evaluate these expressions for given values.

Lesson Notes

In this lesson, students begin by filling in data tables to help them organize data and see patterns; then, they move to

drawing their own tables. Encourage students to label the columns completely.

Classwork

Fluency Exercise (5 minutes): Subtraction of Decimals

Sprint: Refer to the Sprints and Sprint Delivery Script sections in the Module Overview for directions on how to

administer a Sprint.

Opening Exercise (3 minutes)

Opening Exercise

My older sister is exactly two years older than I am. Sharing a birthday is both fun and annoying. Every year on our

birthday, we have a party, which is fun, but she always brags that she is two years older than I am, which is annoying.

Shown below is a table of our ages, starting when I was born:

My Age (in years) My Sister’s Age (in years)

𝟎 𝟐

𝟏 𝟑

𝟐 𝟒

𝟑 𝟓

𝟒 𝟔

Make sure students understand the context of the story problem. It should be clear that the day I was born was my

sister’s second birthday. My first birthday was her third birthday; my second birthday was her fourth birthday; and so

Scaffolding:

Some students benefit from having blank tables prepared ahead of time.

Discussion (5 minutes)

Today in class, we will use data tables. They help us organize data and see patterns. We can use variables to

make generalizations about the patterns we see.

a. Looking at the table, what patterns do you see? Tell a partner.

My sister’s age is always two years more than my age.

b. On the day I turned 𝟖 years old, how old was my sister?

𝟏𝟎 years old

c. How do you know?

Since my sister’s age is always two years more than my age, we just add 𝟐 to my age. 𝟖 + 𝟐 = 𝟏𝟎

d. On the day I turned 𝟏𝟔 years old, how old was my sister?

𝟏𝟖 years old

e. How do you know?

Since my sister’s age is always two years more than my age, we just add 𝟐 to my age. 𝟏𝟔 + 𝟐 = 𝟏𝟖

f. Do we need to extend the table to calculate these answers?

No; the pattern is to add 𝟐 to your age to calculate your sister’s age.

Example 1 (5 minutes)

Example 1

𝟎 𝟐

𝟏 𝟑

𝟐 𝟒

𝟑 𝟓

𝟒 𝟔

𝒀 𝒀 + 𝟐

a. What if you don’t know how old I am? Let’s use a variable for my age. Let 𝒀= my age in years. Can you

develop an expression to describe how old my sister is?

Your sister is 𝒀 + 𝟐 years old.

b. Please add that to the last row of the table.

My age is 𝑌 years. My sister is 𝑌 + 2 years old. So, no matter what my age is (or was), my sister’s age in years

will always be two years greater than mine.

Scaffolding:

A number line in the classroom

can provide an additional

reference for students. A

cardboard sheet with two

windows cut to reveal 𝑌 and

𝑌 + 2 works with Examples 1

and 2.

Example 2 (5 minutes)

Example 2

𝟎 𝟐

𝟏 𝟑

𝟐 𝟒

𝟑 𝟓

𝟒 𝟔

𝑮 − 𝟐 𝑮

a. How old was I when my sister was 𝟔 years old?

𝟒 years old

b. How old was I when my sister was 𝟏𝟓 years old?

𝟏𝟑 years old

c. How do you know?

My age is always 𝟐 years less than my sister’s age.

d. Look at the table in Example 2. If you know my sister’s age, can you determine my age?

We can subtract two from your sister’s age, and that will equal your age.

e. If we use the variable 𝑮 for my sister’s age in years, what expression would describe my age in years?

𝑮 − 𝟐

f. Fill in the last row of the table with the expressions.

My age is 𝑮 − 𝟐 years. My sister is 𝑮 years old.

g. With a partner, calculate how old I was when my sister was 𝟐𝟐, 𝟐𝟑, and 𝟐𝟒 years old.

You were 𝟐𝟎, 𝟐𝟏, and 𝟐𝟐 years old, respectively.

Exercises (15 minutes)

Exercises

1. Noah and Carter are collecting box tops for their school. They each bring in 𝟏 box top per day starting on the first

day of school. However, Carter had a head start because his aunt sent him 𝟏𝟓 box tops before school began.

Noah’s grandma saved 𝟏𝟎 box tops, and Noah added those on his first day.

a. Fill in the missing values that indicate the total number of box tops each boy brought to school.

School Day Number of Box Tops

Noah Has

Number of Box Tops

Carter Has

𝟏 𝟏𝟏 𝟏𝟔

𝟐 𝟏𝟐 𝟏𝟕

𝟑 𝟏𝟑 𝟏𝟖

𝟒 𝟏𝟒 𝟏𝟗

𝟓 𝟏𝟓 𝟐𝟎

b. If we let 𝑫 be the number of days since the new school year began, on day 𝑫 of school, how many box tops

will Noah have brought to school?

𝑫 + 𝟏𝟎 box tops

c. On day 𝑫 of school, how many box tops will Carter have brought to school?

𝑫 + 𝟏𝟓 box tops

d. On day 𝟏𝟎 of school, how many box tops will Noah have brought to school?

𝟏𝟎 + 𝟏𝟎 = 𝟐𝟎; On day 𝟏𝟎, Noah would have brought in 𝟐𝟎 box tops.

e. On day 𝟏𝟎 of school, how many box tops will Carter have brought to school?

𝟏𝟎 + 𝟏𝟓 = 𝟐𝟓; On day 𝟏𝟎, Carter would have brought in 𝟐𝟓 box tops.

2. Each week the Primary School recycles 𝟐𝟎𝟎 pounds of paper. The Intermediate School also recycles the same

amount but had another 𝟑𝟎𝟎 pounds left over from summer school. The Intermediate School custodian added this

extra 𝟑𝟎𝟎 pounds to the first recycle week.

a. Number the weeks, and record the amount of paper recycled by both schools.

Week Total Amount of Paper Recycled by the

Primary School This School Year in Pounds

Total Amount of Paper Recycled by the

Intermediate School This School Year in Pounds

𝟏 𝟐𝟎𝟎 𝟓𝟎𝟎

𝟐 𝟒𝟎𝟎 𝟕𝟎𝟎

𝟑 𝟔𝟎𝟎 𝟗𝟎𝟎

𝟒 𝟖𝟎𝟎 𝟏, 𝟏𝟎𝟎

𝟓 𝟏, 𝟎𝟎𝟎 𝟏, 𝟑𝟎𝟎

b. If this trend continues, what will be the total amount collected for each school on Week 𝟏𝟎?

The Primary School will have collected 𝟐, 𝟎𝟎𝟎 pounds. The Intermediate School will have collected

𝟐, 𝟑𝟎𝟎 pounds.

3. Shelly and Kristen share a birthday, but Shelly is 𝟓 years older.

a. Make a table showing their ages every year, beginning when Kristen was born.

Kristen’s Age (in years) Shelly’s Age (in years)

𝟎 𝟓

𝟏 𝟔

𝟐 𝟕

𝟑 𝟖

b. If Kristen is 𝟏𝟔 years old, how old is Shelly?

If Kristen is 𝟏𝟔 years old, Shelly is 𝟐𝟏 years old.

c. If Kristen is 𝑲 years old, how old is Shelly?

If Kristen is 𝑲 years old, Shelly is 𝑲 + 𝟓 years old.

d. If Shelly is 𝑺 years old, how old is Kristen?

If Shelly is 𝑺 years old, Kristen is 𝑺 − 𝟓 years old.

Closing (2 minutes)

Why were we able to write these expressions?

There was a relationship between the two quantities that we could identify.

What is important to remember about labeling columns in a table?

The label should be complete, with units, so the reader understands precisely what is meant.

How are addition and subtraction expressions related to one another?

They are inverse operations. One undoes the other.

Exit Ticket (5 minutes)

Name Date

Lesson 19: Substituting to Evaluate Addition and Subtraction

Expressions

Exit Ticket

Jenna and Allie work together at a piano factory. They both were hired on January 3, but Jenna was hired in 2005, and

Allie was hired in 2009.

a. Fill in the table below to summarize the two workers’ experience totals.

Year Allie’s Years of Experience Jenna’s Years of Experience

b. If both workers continue working at the piano factory, when Allie has 𝐴 years of experience on the job, how

many years of experience will Jenna have on the job?

c. If both workers continue working at the piano factory, when Allie has 20 years of experience on the job, how

Exit Ticket Sample Solutions

Jenna and Allie work together at a piano factory. They both were hired on January 3, but Jenna was hired in 2005, and

Allie was hired in 2009.

a. Fill in the table below to summarize the two workers’ experience totals.

Year Allie’s Years of Experience Jenna’s Years of Experience

2010 𝟏 𝟓

2011 𝟐 𝟔

2012 𝟑 𝟕

2013 𝟒 𝟖

2014 𝟓 𝟗

b. If both workers continue working at the piano factory, when Allie has 𝑨 years of experience on the job, how

Jenna will have been on the job for 𝑨 + 𝟒 years.

c. If both workers continue working at the piano factory, when Allie has 𝟐𝟎 years of experience on the job, how

𝟐𝟎 + 𝟒 = 𝟐𝟒

Jenna will have been on the job for 𝟐𝟒 years.

Problem Set Sample Solutions

1. Suellen and Tara are in sixth grade, and both take dance lessons at Twinkle Toes Dance Studio. This is Suellen’s first

year, while this is Tara’s fifth year of dance lessons. Both girls plan to continue taking lessons throughout high

school.

a. Complete the table showing the number of years the girls will have danced at the studio.

Grade Suellen’s Years of Experience Dancing Tara’s Years of Experience Dancing

Sixth 𝟏 𝟓

Seventh 𝟐 𝟔

Eighth 𝟑 𝟕

Ninth 𝟒 𝟖

Tenth 𝟓 𝟗

Eleventh 𝟔 𝟏𝟎

Twelfth 𝟕 𝟏𝟏

b. If Suellen has been taking dance lessons for 𝒀 years, how many years has Tara been taking lessons?

Tara has been taking dance lessons for 𝒀 + 𝟒 years.

2. Daejoy and Damian collect fossils. Before they went on a fossil-hunting trip, Daejoy had 𝟐𝟓 fossils in her collection,

and Damian had 𝟏𝟔 fossils in his collection. On a 𝟏𝟎-day fossil-hunting trip, they each collected 𝟐 new fossils each

a. Make a table showing how many fossils each person had in their collection at the end of each day.

Day Number of Fossils in Daejoy’s Collection Number of Fossils in Damian’s Collection

𝟏 𝟐𝟕 𝟏𝟖

𝟐 𝟐𝟗 𝟐𝟎

𝟑 𝟑𝟏 𝟐𝟐

𝟒 𝟑𝟑 𝟐𝟒

𝟓 𝟑𝟓 𝟐𝟔

𝟔 𝟑𝟕 𝟐𝟖

𝟕 𝟑𝟗 𝟑𝟎

𝟖 𝟒𝟏 𝟑𝟐

𝟗 𝟒𝟑 𝟑𝟒

𝟏𝟎 𝟒𝟓 𝟑𝟔

b. If this pattern of fossil finding continues, how many fossils does Damian have when Daejoy has 𝑭 fossils?

When Daejoy has 𝑭 fossils, Damian has 𝑭 − 𝟗 fossils.

c. If this pattern of fossil finding continues, how many fossils does Damian have when Daejoy has 𝟓𝟓 fossils?

𝟓𝟓 − 𝟗 = 𝟒𝟔

When Daejoy has 𝟓𝟓 fossils, Damian has 𝟒𝟔 fossils.

3. A train consists of three types of cars: box cars, an engine, and a caboose. The relationship among the types of cars

is demonstrated in the table below.

Number of Box Cars Number of Cars in the Train

𝟎 𝟐

𝟏 𝟑

𝟐 𝟒

𝟏𝟎 𝟏𝟐

𝟏𝟎𝟎 𝟏𝟎𝟐

a. Tom wrote an expression for the relationship depicted in the table as 𝑩 + 𝟐. Theresa wrote an expression for

the same relationship as 𝑪 − 𝟐. Is it possible to have two different expressions to represent one relationship?

Explain.

Both expressions can represent the same relationship, depending on the point of view. The expression 𝑩 + 𝟐

represents the number of box cars plus an engine and a caboose. The expression 𝑪 − 𝟐 represents the whole

car length of the train, less the engine and caboose.

b. What do you think the variable in each student’s expression represents? How would you define them?

The variable 𝑪 would represent the total cars in the train. The variable 𝑩 would represent the number of box

4. David was 𝟑 when Marieka was born. Complete the table.

Marieka’s Age in Years David’s Age in Years

𝟓 𝟖

𝟔 𝟗

𝟕 𝟏𝟎

𝟖 𝟏𝟏

𝟏𝟎 𝟏𝟑

𝟏𝟕 𝟐𝟎

𝟑𝟐 𝟑𝟓

𝑴 𝑴 + 𝟑

𝑫 − 𝟑 𝑫

5. Caitlin and Michael are playing a card game. In the first round, Caitlin scored 𝟐𝟎𝟎 points, and Michael scored 𝟏𝟕𝟓

points. In each of the next few rounds, they each scored 𝟓𝟎 points. Their score sheet is below.

Caitlin’s Points Michael’s Points

𝟐𝟎𝟎 𝟏𝟕𝟓

𝟐𝟓𝟎 𝟐𝟐𝟓

𝟑𝟎𝟎 𝟐𝟕𝟓

𝟑𝟓𝟎 𝟑𝟐𝟓

a. If this trend continues, how many points will Michael have when Caitlin has 𝟔𝟎𝟎 points?

𝟔𝟎𝟎 − 𝟐𝟓 = 𝟓𝟕𝟓

Michael will have 𝟓𝟕𝟓 points.

b. If this trend continues, how many points will Michael have when Caitlin has 𝑪 points?

Michael will have 𝑪 − 𝟐𝟓 points.

c. If this trend continues, how many points will Caitlin have when Michael has 𝟗𝟕𝟓 points?

𝟗𝟕𝟓 + 𝟐𝟓 = 𝟏𝟎𝟎𝟎

Caitlin will have 𝟏, 𝟎𝟎𝟎 points.

d. If this trend continues, how many points will Caitlin have when Michael has 𝑴 points?

Caitlin will have 𝑴 + 𝟐𝟓 points.

6. The high school marching band has 𝟏𝟓 drummers this year. The band director insists that there are to be 𝟓 more

trumpet players than drummers at all times.

a. How many trumpet players are in the marching band this year?

𝟏𝟓 + 𝟓 = 𝟐𝟎 . There are 𝟐𝟎 trumpet players this year.

b. Write an expression that describes the relationship of the number of trumpet players (𝑻) and the number of

drummers (𝑫).

𝑻 = 𝑫 + 𝟓 or 𝑫 = 𝑻 − 𝟓

c. If there are only 𝟏𝟒 trumpet players interested in joining the marching band next year, how many drummers

will the band director want in the band?

𝟏𝟒 − 𝟓 = 𝟗.

The band director will want 𝟗 drummers.

Subtraction of Decimals—Round 1

Directions: Evaluate each expression.

1. 55 − 50 23. 9.9 − 5

2. 55 − 5 24. 9.9 − 0.5

3. 5.5 − 5 25. 0.99 − 0.5

4. 5.5 − 0.5 26. 0.99 − 0.05

5. 88 − 80 27. 4.7 − 2

6. 88 − 8 28. 4.7 − 0.2

7. 8.8 − 8 29. 0.47 − 0.2

8. 8.8 − 0.8 30. 0.47 − 0.02

9. 33 − 30 31. 8.4 − 1

10. 33 − 3 32. 8.4 − 0.1

11. 3.3 − 3 33. 0.84 − 0.1

12. 1 − 0.3 34. 7.2 − 5

13. 1 − 0.03 35. 7.2 − 0.5

14. 1 − 0.003 36. 0.72 − 0.5

15. 0.1 − 0.03 37. 0.72 − 0.05

16. 4 − 0.8 38. 8.6 − 7

17. 4 − 0.08 39. 8.6 − 0.7

18. 4 − 0.008 40. 0.86 − 0.7

19. 0.4 − 0.08 41. 0.86 − 0.07

20. 9 − 0.4 42. 5.1 − 4

21. 9 − 0.04 43. 5.1 − 0.4

22. 9 − 0.004 44. 0.51 − 0.4

Number Correct: ______ Number Correct: ______

Subtraction of Decimals—Round 1 [KEY]

1. 55 − 50 𝟓 23. 9.9 − 5 𝟒. 𝟗

2. 55 − 5 𝟓𝟎 24. 9.9 − 0.5 𝟗. 𝟒

3. 5.5 − 5 𝟎. 𝟓 25. 0.99 − 0.5 𝟎. 𝟒𝟗

4. 5.5 − 0.5 𝟓 26. 0.99 − 0.05 𝟎. 𝟗𝟒

5. 88 − 80 𝟖 27. 4.7 − 2 𝟐. 𝟕

6. 88 − 8 𝟖𝟎 28. 4.7 − 0.2 𝟒. 𝟓

7. 8.8 − 8 𝟎. 𝟖 29. 0.47 − 0.2 𝟎. 𝟐𝟕

8. 8.8 − 0.8 𝟖 30. 0.47 − 0.02 𝟎. 𝟒𝟓

9. 33 − 30 𝟑 31. 8.4 − 1 𝟕. 𝟒

10. 33 − 3 𝟑𝟎 32. 8.4 − 0.1 𝟖. 𝟑

11. 3.3 − 3 𝟎. 𝟑 33. 0.84 − 0.1 𝟎. 𝟕𝟒

12. 1 − 0.3 𝟎. 𝟕 34. 7.2 − 5 𝟐. 𝟐

13. 1 − 0.03 𝟎. 𝟗𝟕 35. 7.2 − 0.5 𝟔. 𝟕

14. 1 − 0.003 𝟎. 𝟗𝟗𝟕 36. 0.72 − 0.5 𝟎. 𝟐𝟐

15. 0.1 − 0.03 𝟎. 𝟎𝟕 37. 0.72 − 0.05 𝟎. 𝟔𝟕

16. 4 − 0.8 𝟑. 𝟐 38. 8.6 − 7 𝟏. 𝟔

17. 4 − 0.08 𝟑. 𝟗𝟐 39. 8.6 − 0.7 𝟕. 𝟗

18. 4 − 0.008 𝟑. 𝟗𝟗𝟐 40. 0.86 − 0.7 𝟎. 𝟏𝟔

19. 0.4 − 0.08 𝟎. 𝟑𝟐 41. 0.86 − 0.07 𝟎. 𝟕𝟗

20. 9 − 0.4 𝟖. 𝟔 42. 5.1 − 4 𝟏. 𝟏

21. 9 − 0.04 𝟖. 𝟗𝟔 43. 5.1 − 0.4 𝟒. 𝟕

22. 9 − 0.004 𝟖. 𝟗𝟗𝟔 44. 0.51 − 0.4 𝟎. 𝟏𝟏

Subtraction of Decimals—Round 2

1. 66 − 60 23. 6.8 − 4

2. 66 − 6 24. 6.8 − 0.4

3. 6.6 − 6 25. 0.68 − 0.4

4. 6.6 − 0.6 26. 0.68 − 0.04

5. 99 − 90 27. 7.3 − 1

6. 99 − 9 28. 7.3 − 0.1

7. 9.9 − 9 29. 0.73 − 0.1

8. 9.9 − 0.9 30. 0.73 − 0.01

9. 22 − 20 31. 9.5 − 2

10. 22 − 2 32. 9.5 − 0.2

11. 2.2 − 2 33. 0.95 − 0.2

12. 3 − 0.4 34. 8.3 − 5

13. 3 − 0.04 35. 8.3 − 0.5

14. 3 − 0.004 36. 0.83 − 0.5

15. 0.3 − 0.04 37. 0.83 − 0.05

16. 8 − 0.2 38. 7.2 − 4

17. 8 − 0.02 39. 7.2 − 0.4

18. 8 − 0.002 40. 0.72 − 0.4

19. 0.8 − 0.02 41. 0.72 − 0.04

20. 5 − 0.1 42. 9.3 − 7

21. 5 − 0.01 43. 9.3 − 0.7

22. 5 − 0.001 44. 0.93 − 0.7

Number Correct: ______

Improvement: ______

Subtraction of Decimals—Round 2 [KEY]

1. 66 − 60 𝟔 23. 6.8 − 4 𝟐. 𝟖

2. 66 − 6 𝟔𝟎 24. 6.8 − 0.4 𝟔. 𝟒

3. 6.6 − 6 𝟎. 𝟔 25. 0.68 − 0.4 𝟎. 𝟐𝟖

4. 6.6 − 0.6 𝟔 26. 0.68 − 0.04 𝟎. 𝟔𝟒

5. 99 − 90 𝟗 27. 7.3 − 1 𝟔. 𝟑

6. 99 − 9 𝟗𝟎 28. 7.3 − 0.1 𝟕. 𝟐

7. 9.9 − 9 𝟎. 𝟗 29. 0.73 − 0.1 𝟎. 𝟔𝟑

8. 9.9 − 0.9 𝟗 30. 0.73 − 0.01 𝟎. 𝟕𝟐

9. 22 − 20 𝟐 31. 9.5 − 2 𝟕. 𝟓

10. 22 − 2 𝟐𝟎 32. 9.5 − 0.2 𝟗. 𝟑

11. 2.2 − 2 𝟎. 𝟐 33. 0.95 − 0.2 𝟎. 𝟕𝟓

12. 3 − 0.4 𝟐. 𝟔 34. 8.3 − 5 𝟑. 𝟑

13. 3 − 0.04 𝟐. 𝟗𝟔 35. 8.3 − 0.5 𝟕. 𝟖

14. 3 − 0.004 𝟐. 𝟗𝟗𝟔 36. 0.83 − 0.5 𝟎. 𝟑𝟑

15. 0.3 − 0.04 𝟎. 𝟐𝟔 37. 0.83 − 0.05 𝟎. 𝟕𝟖

16. 8 − 0.2 𝟕. 𝟖 38. 7.2 − 4 𝟑. 𝟐

17. 8 − 0.02 𝟕. 𝟗𝟖 39. 7.2 − 0.4 𝟔. 𝟖

18. 8 − 0.002 𝟕. 𝟗𝟗𝟖 40. 0.72 − 0.4 𝟎. 𝟑𝟐

19. 0.8 − 0.02 𝟎. 𝟕𝟖 41. 0.72 − 0.04 𝟎. 𝟔𝟖

20. 5 − 0.1 𝟒. 𝟗 42. 9.3 − 7 𝟐. 𝟑

21. 5 − 0.01 𝟒. 𝟗𝟗 43. 9.3 − 0.7 𝟖. 𝟔

22. 5 − 0.001 𝟒. 𝟗𝟗𝟗 44. 0.93 − 0.7 𝟎. 𝟐𝟑

Lesson 19: Substituting to Evaluate Addition and ...

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